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Intermediate Algebra
Sequences and Series

1
Sequences and Recursion Formulas

Problem  1
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Find the first \(4\) terms of the sequence whose general term is \(a_n=2n-1\)

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Mr. Perez
Mr. Perez
Betsy cc
Betsy
Brooke
Brooke
Edwin espanol spanish
Edwin
Problem  2
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Find the first \(4\) terms of the sequence whose general term is \(a_n=\displaystyle\frac{1}{n+1}\)

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Mr. Perez
Mr. Perez
Stefanie cc
Stefanie
Brooke
Brooke
Cynthia espanol spanish
Cynthia
Problem  3
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Find the \(5^{\text{th}}\) and \(6^{\text{th}}\) terms of the sequence whose general term is \(a_n=\displaystyle\frac{(-1)^n}{n^2}\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  4
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Find the first \(4\) terms of the sequence given recursively by \(a_1=4\) and \(a_n=5a_{n-1}\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  5

Find the formula for the \(n\)th term for \(2, 8, 18, 32, \ldots\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  6
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Find the general term for \(2, \displaystyle\frac{3}{8}, \displaystyle\frac{4}{27}, \displaystyle\frac{5}{64}, \ldots\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia

2
Series

Problem  1
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Expand and simplify \(\displaystyle\sum_{i=1}^{5} \left(i^2-1\right)\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  2
practice icon

Expand and simplify \(\displaystyle\sum_{i=3}^{6}(-2)^i\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  3
practice icon

Expand \(\displaystyle\sum_{i=2}^{5}\left(x^i-3\right)\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  4
practice icon

Write with summation notation \(1+3+5+7+9\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  5
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Write with summation notation \(3+12+27+48\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  6
practice icon

Write with summation notation \(\displaystyle\frac{x+3}{x^3}+ \displaystyle\frac{x+4}{x^4}+ \displaystyle\frac{x+5}{x^5}+ \displaystyle\frac{x+6}{x^6}\)

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Julieta cc
Julieta
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Mini Lecture
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Mini Lecture
Expand and simplify.

  1. \(\displaystyle\sum_{i=1}^{4}(2t+4)\)

  2. \(\displaystyle\sum_{i=3}^{6}(-2)^i\)

  3. \(\displaystyle\sum_{i=3}^{6}(x+i)^i\)

Write with summation notation.

  1. \(\displaystyle\frac{3}{4}+\displaystyle\frac{4}{5}+\displaystyle\frac{5}{6}+\displaystyle\frac{6}{7}+\displaystyle\frac{7}{8}\)

  2. \(4+8+16+32+64\)

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Mr. McKeague cc
Mr. McKeague

3
Arithmetic Sequences

Problem  1
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Give the common difference \(d\) for the arithmetic sequence \(4, 10, 16, 22, \ldots\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  2
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Find the common difference for \(100, 93, 86, 79, \ldots\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  3
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Find the common difference for \(\displaystyle\frac{1}{2}, 1, \displaystyle\frac{3}{2}, 2, \ldots\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  4
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Find the general term for \(7, 19, 13, 16, \ldots\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  5

Find the general term of the arithmetic progression whose \(3^{\text{rd}}\) term \(a_3\) is \(7\) and \(8^{\text{th}}\) term \(a_8\) is \(17\).

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  6
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Find the sum of the first \(10\) terms of \(2, 10, 18, 26, \ldots\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Mini Lecture
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Mini Lecture
Is the sequence arithmetic?

  1. \(50, 45, 40, \ldots\)

  2. \(1, 4, 9, 16, \ldots\)

  3. If \(a_1=3\) and \(d=4\), find \(a_n\) and \(a_{24}\)

  4. If \(a_6=17\) and \(a_{12}=29\), find \(a_1\), \(d\), and \(a_{30}\)

  5. Find \(S_{100}\) for \(5, 9, 13, 17, \ldots\)

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Mr. McKeague cc
Mr. McKeague

4
Geometric Sequences

Problem  1
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Find the common ratio for \(\displaystyle\frac{1}{2}, \displaystyle\frac{1}{4}, \displaystyle\frac{1}{8}, \displaystyle\frac{1}{16}, \ldots\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  2
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Find the common ratio for \(\sqrt{3}, 3, 3\sqrt{3}, 9, \ldots\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  3
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Find the general term for \(5, 10, 20, \ldots\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  4
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Find the tenth term of the sequence \(3, \displaystyle\frac{3}{2}, \displaystyle\frac{3}{4}, \displaystyle\frac{3}{8}, \ldots\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  5

Find the general term of the geometric progression whose \(4^{\text{th}}\) term is \(16\) and whose \(7^{\text{th}}\) term is \(128\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  6
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Find the sum of the first \(10\) terms of \(5, 15, 45, 135, \ldots\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  7
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Find the sum of the infinite series \(\displaystyle\frac{1}{5}+ \displaystyle\frac{1}{10}+ \displaystyle\frac{1}{20}+ \displaystyle\frac{1}{40}+ \ldots\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  8

Show that \(0.999\ldots\) is equal to \(1\).

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Stefanie cc
Stefanie
CJ cc
CJ
Julieta espanol spanish
Julieta
Problem  9
practice icon

Mini Lecture
Is the sequence geometric?

  1. \(1, 5, 25, 125, \ldots\)

  2. \(\displaystyle\frac{1}{2}, \displaystyle\frac{1}{6}, \displaystyle\frac{1}{18}, \displaystyle\frac{1}{54}, \ldots\)

  3. If \(a_1=4\) and \(r=3\), find \(a_n, a_{20},\) and \(S_{20}\)

  4. Find \(a_{10}\) and \(S_{10}\) for \(\sqrt{2}, 2, 2\sqrt{2}, \ldots\) \(\displaystyle\frac{1}{2}+\displaystyle\frac{1}{4}+\displaystyle\frac{1}{8}+\ldots=\)?

  5. Show that \(0.444\ldots=\displaystyle\frac{1}{9}\)

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Mr. McKeague cc
Mr. McKeague

5
The Binomial Expansion

Problem  1

Calculate the following binomial coefficients: \(\displaystyle\binom{7}{5}\), \(\displaystyle\binom{6}{2}\), \(\displaystyle\binom{3}{0}\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  2
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Expand \((x-2)^3\)

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Breylor cc
Breylor
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  3
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Expand \((3x+2y)^4\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  4
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Find the first three terms in the expansion of \((x+5)^9\)

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Stefanie cc
Stefanie
CJ cc
CJ
Cynthia espanol spanish
Cynthia
Problem  5
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Find the fifth term in the expansion of \((2x+3y)^{12}\)

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Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  6

Mini Lecture
Expand.

  1. \((x+2)^4\)

  2. \((4x-3y)^3\)

  3. Write the first four terms: \((x-y)^{10}\)

  4. Write the first two terms: \((x+2)^{100}\)

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Mr. McKeague cc
Mr. McKeague

6
Venn Diagrams, Permutations, and Combinations

Problem  1

Suppose a sample space is a deck of \(52\) playing cards. Let set \(A=\{ \text{Aces} \}\) and \(B=\{ \text{Kings} \}\) and use a Venn diagram to show that \(A\) and \(B\) are mutually exclusive.

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Mr. Perez
Mr. Perez
Joshua cc
Joshua
Stephanie cc
Stephanie
Julieta cc espanol spanish
Julieta
Problem  2

Use a Venn diagram to show the intersection of the set \(A=\{\text{Aces}\}\) and \(B=\{\text{Spades}\}\) from the sample space of a deck of playing cards.

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Mr. Perez
Mr. Perez
Logan cc
Logan
Stephanie cc
Stephanie
Julieta cc espanol spanish
Julieta
Problem  3
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Use Venn diagrams to check the following expression. \[A\cap(B\cup C)=(A\cap B)\cup (A\cap C)\]

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Mr. McKeague cc
Mr. McKeague
Julieta cc
Julieta
Logan cc
Logan
Mr. Perez
Mr. Perez
Problem  4
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Let \(A\) and \(B\) be two intersecting sets, neither of which is a subset of the other. Use a Venn diagram to illustrate the set \[\{x\mid x\in A\text{ and }x\notin B\}\]

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Mr. McKeague cc
Mr. McKeague
Stephanie cc
Stephanie
Mr. Perez
Mr. Perez
Julieta espanol spanish
Julieta
Problem  5

Find the number of permutations of the letters in the set \(\{\text{E, N, G, L, I, S, H}\}\).

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Joshua cc
Joshua
Stephanie cc
Stephanie
Julieta cc espanol spanish
Julieta
Problem  6
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How many different ways are there to arrange the letters in the word CHEMISTRY?

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Logan cc
Logan
Stephanie cc
Stephanie
Julieta cc espanol spanish
Julieta
Problem  7

How many strings of letters can be made from the letters in the word FINAL, if no letter can be repeated?

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Mr. McKeague cc
Mr. McKeague
Logan cc
Logan
Julieta cc espanol spanish
Julieta
Problem  8
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In how many ways can \(8\) people fill \(5\) chairs at a table?

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Joshua cc
Joshua
Stephanie cc
Stephanie
Julieta espanol spanish
Julieta
Problem  9

How many ways are there for three people from the set \(\{\text{Pat, Diane, Tim, JoAnn}\}\) to win first, second, and third prize in a contest?

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Julieta cc
Julieta
Julieta cc espanol spanish
Julieta
Problem  10
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How many three-person committees can be formed from people in the set \(\{\text{Pat, Diane, Tim, JoAnn}\}\)?

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Mr. McKeague cc
Mr. McKeague
Stephanie cc
Stephanie
Julieta espanol spanish
Julieta
Problem  11

How many four-person committees can be selected from a set of \(7\) people?

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Mr. McKeague cc
Mr. McKeague
Stephanie cc
Stephanie
Julieta espanol spanish
Julieta
Problem  12
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A jar contains \(4\) coins: a penny, a nickel, a dime, and a quarter. If \(2\) coins are selected, how many different amounts of money are possible?

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Mr. McKeague cc
Mr. McKeague
Stephanie cc
Stephanie
Julieta espanol spanish
Julieta